Question number 22. The figure shown here and my enlarged version has the following properties. PQR is a rectangle and T is the midpoint of RS. So let's begin right away by labeling these segments with information we know. If T is the midpoint then ST and TR have the same length. If PQRS is a rectangle then we have some right angles over here in the corners at SPQ and R. Then we also know that QT is perpendicular to the diagonal so here we have another right angle and let me call the point of intersection H. And our job is to find the ratio PQ to QR. So we will be using similar triangles to solve the problem. Let's label the angles that we know are the same by alternate interior angles. These angles are the same. Again by alternate interior angles we have another pair of equal angles here and now that tells us that we also have the same angles here. So we have three similar right triangles in this picture. We also know something about the length of the segment TR. We can label TR here as exactly equal to 1 half PQ like so because T is the midpoint of SR. Okay so we can now use similar triangles. We have that triangle PQR is similar to triangle QRT and that will give us the following ratio so PQ is to QR as QR is to RT and then we know that RT is exactly 1 half PQ. So we can use this relationship now with the initial above equation that we have. So let me go up there and write that PQ is to QR as QR is to 1 half of PQ and then we cross multiply we have that 1 half now this is going to be PQ that length squared equal to QR squared and then we are looking for the ratio of PQ to QR so I will divide by a half on both sides we have PQ quantity squared is 2 QR quantity squared I will take a square root of both sides so PQ is equal to the square root of times QR like that and then finally PQ over QR is going to be square root of 2 and over 1 so the ratio is root 2 to 1 using similar triangles and that is answer D.

rectangle

similar triangles

ratio

midpoint

geometry